Question

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for each triangle, check all that apply.
triangle a
triangle b
triangle c
triangle d
(figures of four triangles: triangle a is a right triangle with angles 90°, 60°, 30°; triangle b has all sides 7; triangle c has sides 5, 7, 11; triangle d is a right triangle with two 45° angles)
scalene
isosceles
equilateral
scalene
isosceles
equilateral
scalene
isosceles
equilateral
scalene
isosceles
equilateral
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Explanation:

for Each Triangle:

Triangle A:

• A right triangle with angles \(90^\circ\), \(60^\circ\), \(30^\circ\). The sides will be in the ratio \(1:\sqrt{3}:2\) (all different), so it's scalene. Also, since it has a right angle, but let's check the type by sides: scalene (all sides different), not isosceles (no two sides equal), not equilateral (all sides equal).

Triangle B:

• All sides are 7, so it's equilateral (which is a special case of isosceles). So it's isosceles (at least two sides equal) and equilateral (all three sides equal).

Triangle C:

• Sides are 5, 7, 11 (all different), so it's scalene. Not isosceles (no two sides equal), not equilateral (all sides equal).

Triangle D:

• Angles are \(90^\circ\), \(45^\circ\), \(45^\circ\). So two angles are equal, meaning two sides are equal (isosceles). Sides: the legs are equal (since angles opposite are equal), hypotenuse is different. So isosceles (not scalene, not equilateral).

Step - by - Step Classification:

Triangle A:

• Check Scalene: Yes (all sides different as per angle - side relationship in right - triangle \(30 - 60 - 90\)).
• Check Isosceles: No (no two sides equal).
• Check Equilateral: No (all sides not equal).

Triangle B:

• Check Scalene: No (all sides equal).
• Check Isosceles: Yes (at least two sides equal, in fact all three).
• Check Equilateral: Yes (all three sides equal).

Triangle C:

• Check Scalene: Yes (sides 5, 7, 11 are all different).
• Check Isosceles: No (no two sides equal).
• Check Equilateral: No (all sides not equal).

Triangle D:

• Check Scalene: No (two sides equal as two angles are \(45^\circ\)).
• Check Isosceles: Yes (two sides equal).
• Check Equilateral: No (all sides not equal, one side is hypotenuse different from legs).

Answer:

(Checkboxes):

• Triangle A: \(\boxed{\text{Scalene}}\)
• Triangle B: \(\boxed{\text{Isosceles}}\), \(\boxed{\text{Equilateral}}\)
• Triangle C: \(\boxed{\text{Scalene}}\)
• Triangle D: \(\boxed{\text{Isosceles}}\)